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The SABR Model: Smile Dynamics, Backbones and Hagan's Formula
A six-part series on how option markets price volatility, from the textbook model to the arbitrage-free surfaces used by professional desks.
In 2002, Patrick Hagan and three co-authors published a paper with a provocative message: local volatility, the industry standard at the time, could produce worse hedges than plain Black-Scholes because it predicts the smile moving the wrong way. Their alternative, the SABR model, became the market standard for interest rate options within a few years and remains one of the most widely used smile models in the world.
The model
SABR stands for "stochastic alpha, beta, rho." It describes a forward price F, such as a forward swap rate, a forward FX rate or an equity forward, together with its volatility α:
Compared with Heston, the model is deliberately simple. Volatility has no mean reversion; it simply follows a lognormal random walk with volatility ν. The exponent β, between 0 and 1, controls how the size of price moves depends on the price level: β = 1 gives lognormal behavior like Black-Scholes, and β = 0 gives normal, or absolute, behavior often used for interest rates.
SABR is used one expiration at a time. Each maturity gets its own parameters, which makes it a smile model rather than a full model of how the whole surface evolves.
Four parameters, four jobs
| Parameter | Role | How it is usually set |
|---|---|---|
| α | Overall volatility level | Solved so that the model matches at-the-money volatility exactly |
| β | Backbone: how ATM volatility moves with the forward | Fixed by convention or from historical behavior, often 0, 0.5 or 1 |
| ρ | Correlation between forward and volatility | Fitted to the skew; negative for equity indices |
| ν | Volatility of volatility | Fitted to the curvature of the wings |
This clean division of labor is a large part of SABR's appeal. Traders can read the skew directly from ρ and the wing curvature from ν, and when the market moves they can see which feature of the smile changed.
Hagan's formula
SABR does not have an exact closed-form price, but Hagan and his co-authors derived an accurate approximation for Black-Scholes implied volatility using singular perturbation methods. At the money it reads:
For other strikes the formula is longer, but in the lognormal case β = 1 it becomes compact:
The ratio z / x(z) carries the shape of the smile. When ρ is negative, it rises for strikes below the forward and falls above it, creating a skew; ν controls how quickly it bends away from the at-the-money level.
Getting the dynamics right
The main reason SABR displaced local volatility in many markets is its behavior when the underlying moves. In SABR, when the forward moves, the whole smile moves with it, shifting in the same direction, and the at-the-money level follows a path called the backbone, determined by β. With β = 1, at-the-money volatility stays roughly constant as the forward moves. With β = 0, at-the-money volatility in lognormal terms rises as the forward falls, which matches how interest rate volatility often behaves.
This matters for hedging. The true delta of an option includes the change in its implied volatility when the underlying moves:
A model that predicts the wrong sign for ∂σ / ∂F produces a hedge that is systematically off. SABR's smile dynamics give this term the right sign in markets where the smile tends to move with the forward. The extra term is closely related to the vanna exposure discussed in the GEX guide: it is the part of an option's price sensitivity that comes from volatility changing alongside price.
Where SABR is used
- Interest rates. SABR is the standard way to quote and interpolate smiles for swaptions and caps. Each expiry and swap tenor gets its own parameter set, with β usually fixed across the whole cube.
- FX. It is often used to build smiles from the standard at-the-money, risk reversal and butterfly quotes discussed in Part 2.
- Equities and commodities. It is used for smile interpolation, though equity desks more often rely on local or stochastic local volatility for the full surface.
Limitations and fixes
Hagan's formula is an approximation, and it breaks down in a predictable place: low strikes at long maturities, especially when rates are low. There the implied density can become negative, which is butterfly arbitrage. After rates fell close to zero, and then below it in several currencies, this became a practical problem. The market responded in two ways:
- Shifted SABR applies the model to F + s instead of F, so that slightly negative rates can be handled.
- Arbitrage-free SABR, introduced by Hagan and co-authors in 2014, solves a one-dimensional equation for the density numerically rather than relying on the expansion, guaranteeing a valid distribution while staying close to the original smiles.
The deeper limitation is the one already mentioned: SABR treats each expiration separately and has no mean reversion, so it says little about how different maturities relate to each other. Building a full surface that is consistent across both strikes and maturities is the topic of the final part of this series.
Key takeaways
- SABR models a forward and its volatility with four parameters: α for level, β for the backbone, ρ for skew and ν for curvature.
- Hagan's approximation gives implied volatility directly, which makes calibration fast and transparent.
- The model's smile moves with the underlying, fixing the hedging problem identified in local volatility.
- It is the market standard for interest rate options and widely used in FX.
- The approximation can produce arbitrage at low strikes and long maturities, which shifted and arbitrage-free variants address.